Harmonic Balance FE Magnetic Vector Potential Model Reference

The corresponding solver calculates the complex magnetic vector potential according to Eqn. (4308).

Model Properties

Regularization Parameter

Specifies the regularization parameter $\tilde{\kappa }$ defined in Eqn. (4310).

If the simulation has convergence issues, you can try increasing the regularization parameter progressively, while monitoring the solution. You are advised to keep its value small.

Integration Order Option, Integration Order

Simcenter STAR-CCM+ solves the discretized form of Eqn. (4308) using Gaussian integration rules. These properties allow you to specify the integration order as follows:

Integration Order Option	Integration Order
Auto Simcenter STAR-CCM+ automatically determines an appropriate integration order for the polynomial degree of the shape functions.	Any value that is specified under this property is ignored.
Relative Allows you to increase the integration order that is automatically determined by Simcenter STAR-CCM+ by an additive value.	Allows you to specify a value which Simcenter STAR-CCM+ adds to the automatic integration order.
Absolute Gives you full control on the integration order.	Allows you to specify the integration order as an absolute value.

Reference Values

Excitation Frequency

Allows you to specify the frequency $f_0$ in Eqn. (4245). Simcenter STAR-CCM+ requires the frequency to be the same across contact or baffle interfaces. When you assign different continua to contacting regions, make sure that you set the same frequency in both continua.

Material Properties

Magnetic Permeability

Specifies the magnetic permeability $\mu$ as a scalar profile using either a constant input or a field function. At low-frequency, the magnetic permeability can be assumed to be a real quantity.

Electrical Conductivity

Specifies the electrical conductivity $\sigma$ in Eqn. (4254) as a complex profile. See Eqn. (4263) and [eqnlink].

Initial Conditions

Magnetic Vector Potential

Allows you to initialize the magnetic vector potential $\hat{A}$ (see Eqn. (4254)). Set as a complex profile.

Boundary Settings

Magnetic Vector Potential Specification

Simcenter STAR-CCM+ provides several methods to specify the magnetic vector potential and the electric current sheet at boundaries (see 边界和交界面条件). For harmonic time dependence, these quantities are specified as complex profiles.

Method	Corresponding Physics Value Nodes
Electric Current Sheet Sets the electric current sheet ${\hat{J}}_S$ to the tangential component of a specified complex profile, ${\hat{\overline{J}}}_S$ : $${\hat{J}}_S={\hat{\overline{J}}}_S{|}_{t_1,t_2}$$	Electric Current Sheet Allows you to specify a complex profile, ${\hat{\overline{J}}}_S$ . Simcenter STAR-CCM+ applies the components of ${\hat{\overline{J}}}_S$ tangential to the boundary and neglects the component normal to the boundary. See Eqn. (4260) and Eqn. (4261).
Magnetic Vector Potential Sets the tangential component of the magnetic vector potential $\hat{A}$ to the tangential component of a specified complex profile, $\hat{\overline{A}}$ : $$\hat{A}{|}_{t_1,t_2}=\hat{\overline{A}}{|}_{t_1,t_2}$$	Magnetic Vector Potential Allows you to specify a complex profile, $\hat{\overline{A}}$ . Simcenter STAR-CCM+ applies the components of $\hat{\overline{A}}$ tangential to the boundary, and neglects the normal component. See Eqn. (4256) and Eqn. (4257).
Tangential Magnetic Field Sets the electric current sheet to the tangential component of a specified magnetic field ${\hat{\overline{H}}}_S$ : $${\hat{J}}_S={\hat{\overline{H}}}_S\times n$$ where $n$ is the unit vector normal to the boundary.	Specific Tangential Magnetic Field Allows you to specify the magnetic field ${\hat{\overline{H}}}_S$ as a complex profile. Any normal component of the specified ${\hat{\overline{H}}}_S$ is ignored.
Symmetry - Perfect Magnetic Conductor Sets the electric current sheet to zero: $${\hat{J}}_S=0$$	None
Anti-Symmetry - Perfect Electric Conductor Sets the tangential component of the magnetic vector potential $\hat{A}$ to zero, while leaving the normal component free: $$\begin{array}{l}\hat{A}{|}_{t_1,t_2}=0\\ {\hat{A}}_{n}free\end{array}$$ Most commonly, you specify $\hat{A}{|}_{t_1,t_2}=0$ at a boundary to prevent any magnetic flux from crossing the boundary.	None
Impedance Models the magnetic field ${\widehat{\mathbf{\text{H}}}}_s$ in the vicinity of a conductor without including the conductor interior: $${\widehat{\mathbf{\text{H}}}}_s\times \mathbf{n}=-\frac{i\omega }{{\hat{Z}}_s}\hat{\mathbf{A}}\times \mathbf{n}$$ You can calculate an approximate impedance ${\hat{Z}}_s$ based on the conducting material, using: $${\hat{Z}}_s=\frac{1+i}{\sigma \delta }$$ with a skin depth $\delta$ $$\delta =1/\sqrt{\pi f\mu \sigma }$$ (350) where: $f$ is frequency. $\mu$ is the permeability of the region. $\sigma$ is the conductivity of the region. This approach is particularly suited for cases where $\delta$ is much smaller than the actual thickness of the domain that is modelled but not meshed.	Impedance Allows you to specify the surface impedance ${\hat{Z}}_s$ with a complex profile using standard methods for each component.

Region Settings

Applies to fluid, porous, and solid regions.

Electric Current Density Source Option

Allows you to specify an additional electric current density source for the region ( $J_u$ in Eqn. (4311)).

Method	Corresponding Physics Value Nodes
None No electric current density source is defined.	None
Specified Specifies a user-defined electric current density source ($J_u$ in Eqn. (4311)).	Complex Electric Current Density Source Allows you to specify the electric current density as a complex profile. See Eqn. (4260) and Eqn. (4261).

Mid-side Vertex Option

Allows you to add mid-side vertices to the mesh edges. Mid-side vertices allow for:

• Improved visualization for simulations containing non-tetrahedral mesh elements or a higher-order magnetic vector potential solution (see Finite Element Magnetic Vector Potential Model Properties).
• Improved evaluation of Nodal Forces (see Magnetic Nodal Force Model Reference)
• A conformal interface in cases where the electromagnetic region interfaces a solid region with mid-side vertices (for example, a solid region that requires second-order accuracy for the thermal solution with the Finite Element Solid Energy model).

Use the same Mid-side Vertex Option for all connected solid regions in a simulation. For more information on the methods available, see Mid-side Vertex Option.

Solver Properties

Solver Method

Allows you to choose the type of solver that Simcenter STAR-CCM+ uses to compute the solution.

Method	Corresponding Nodes
Direct Simcenter STAR-CCM+ computes the solution using a sparse direct solver. This is the default option.	Direct For information on the direct solver properties, see FE Sparse Direct Solver Reference. For large cases, a direct solution approach can result in large memory and time requirements, especially for parallel execution. For these cases, you can choose the iterative solution approach.
Iterative Simcenter STAR-CCM+ computes the solution using an iterative solver. This option is appropriate for simulations with large memory requirements and real conductivity.	Iterative For information on the iterative solver properties, see FE Iterative Solver Reference. The Iterative solver has the following controls: Flexible GMRES The Hypre Flexible GMRES solver. See Flexible GMRES Solver Properties. Iterative The preconditioner for the Flexible GMRES solver. The default Convergence Tolerance is 0.01, but the other property settings are as for the Iterative solver. See Iterative Solver Properties and Iterative Solver Controls.

冻结求解器

开启时，求解器在迭代过程中不更新任何物理量。该选项默认情况下关闭。这是一个调试选项，由于缺少储存，它可能导致不可恢复的错误和错误的求解。有关详细信息，请参见有限体积求解器参考。

Field Functions

Electric Current Density (Imag, Real, Phase)

Represent the imaginary part, real part, and phase of the complex electric current density $\hat{J}$ (see Eqn. (4269)).

Electric Current Sheet (Imag, Real, Phase, Magnitude)

Represent the imaginary part, real part, phase, and magnitude of the complex electric current sheet ${\hat{J}}_S$ (see Eqn. (4313)).

Magnetic Field (Imag, Real, Phase)

Represent the imaginary part, real part, and phase of the complex magnetic field $\hat{H}$ (see Eqn. (4268)).

Magnetic Flux Density (Imag, Real, Phase)

Represent the imaginary part, real part, and phase of the complex magnetic flux density $\hat{B}$ (see Eqn. (4268)).

Magnetic Vector Potential (Imag, Real, Phase)

Represent the imaginary part, real part, and phase of the complex magnetic vector potential $\hat{A}$ (see Eqn. (4252)).

Permeability

Represents the magnetic permeability $\mu$ of the material. The magnetic permeability is assumed to be a real quantity.